The phase diagram and the critical behavior of the planar rotator model on square lattice with fourfold and on triangular lattice with sixfold symmetry-breaking field is obtained by Monte Carlo simulation. On the basis of a renormalization-group analysis the planar model with a fourfold symmetry-breaking field is expected to undergo a continuous phase transition with unconventional field dependent critical exponents, while the same model with sixfold symmetry-breaking field is expected to exhibit two phase transitions: a ferromagnetic--BKT (Berezinskii-Kosterlitz-Thouless) one at low temperature followed by a BKT--paramagnetic phase transition at high temperature. Both these expectations are confirmed by Monte Carlo simulations.
The phase diagram and the critical behavior of the planar rotator model on square lattice with fourfold and on triangular lattice with sixfold symmetry-breaking field is obtained by Monte Carlo simulation. On the basis of a renormalization-group analysis the planar model with a fourfold symmetry-breaking field is expected to undergo a continuous phase transition with unconventional field dependent critical exponents, while the same model with sixfold symmetry-breaking field is expected to exhibit two phase transitions: a ferromagnetic--BKT (Berezinskii-Kosterlitz-Thouless) one at low temperature followed by a BKT--paramagnetic phase transition at high temperature. Both these expectations are confirmed by Monte Carlo simulations.
The Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two-dimensional planar rotator and XY models on a square lattice, diluted by randomly placed vacancies, is studied here using hybrid Monte Carlo simulations that combine single spin flip, cluster, and over-relaxation techniques. The transition temperature Tc is determined as a function of vacancy density ${\ensuremath{\rho}}_{\mathrm{vac}}$ by finite-size scaling of the helicity modulus and the in-plane magnetic susceptibility. The results for Tc are consistent with those from the much less precise fourth-order cumulant of Binder. Tc is found to decrease monotonically with increasing ${\ensuremath{\rho}}_{\mathrm{vac}}$, and falls to zero close to the square lattice percolation limit, ${\ensuremath{\rho}}_{\mathrm{vac}}\ensuremath{\approx}0.41$. The result is physically reasonable: the quasi-long-range orientational order of the low-temperature phase cannot be maintained in the absence of sufficient spin interactions across the lattice.
The Berezinskii-Kosterlitz-Thouless (BKT) transition in two-dimensional planar rotator and XY models on a square lattice, diluted by randomly placed vacancies, is studied here using hybrid Monte Carlo simulations that combine single spin flip, cluster and over-relaxation techniques. The transition temperature Tc is determined as a function of vacancy density ρvac by calculations of the helicity modulus and the by finite-size scaling of the in-plane magnetic susceptibility. The results for Tc are consistent with those from the much less precise fourth-order cumulant of Binder. Tc is found to decrease monotonically with increasing ρvac, and falls to zero close to the square lattice percolation limit, ρvac≈0.41 . The result is physically reasonable: the long-range orientational order of the low-temperature phase cannot be maintained in the absence of sufficient spin interactions across the lattice.
Abstract In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii-Kosterlitz-Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various two-dimensional lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases --- namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.
Abstract In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii–Kosterlitz–Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various 2D lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases—namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.
Results of Monte Carlo simulations of the ferromagnetic classical XXZ model on a square lattice are reported for interactions of the out of plane spin components \ensuremath{\lambda} varying between 0 and 0.99. For all the value of \ensuremath{\lambda} the signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition was observed, and the corresponding critical temperature was located both by BKT fits and finite-size scaling methods. Data for in-plane and out of plane susceptibility, correlation length, specific heat, and internal energy are given in a broad range of temperature and for lattice sizes ranging from 32\ifmmode\times\else\texttimes\fi{}32 to 256\ifmmode\times\else\texttimes\fi{}256.
We investigate the isotropic-anisotropic phase transition of the twodimensional XY model with six-fold anisotropy, using Monte Carlo renormalization group method. The result indicates difficulty of observing asymptotic critical behavior in Monte Carlo simulations, owing to the marginal flow at the fixed point. PACS numbers: 02.70.Lq, 75.10.Hk Typeset using REVTEX 1 The XY model with Z6 symmetry breaking field represents not only planar spin magnets with Z6 symmetric crystal field [1], but also models with Z2 × Z3 symmetry (which is isomorphic to Z6) such as three-state antiferromagnetic Potts model on the square and the cubic lattice [2] and Ising antiferromagnet on the triangular and the stacked-triangular lattice [3–8]. The critical behavior of the model in two dimension is well understood by scaling argument [1]: The model undergoes two distinct phase transitions, both of which being